JHU Number Theory Seminar, Fall 2003

Abstract of Talks

Sep. 24, 5:00pm--6:00pm

Gautam Chinta, Nonvanishing Twists of GL(2) L-functions
We describe the method of multiple Dirichlet series and present a nonvanishing result for central values of twists by a character of fixed finite order of an L-function of an automorphic form on GL(2).

Oct. 8, 4:00pm--5:00pm

Yangbo Ye, Selberg's Orthognality for Automorphic L-functions
Selberg's orthogonality conjecture predicts that the coefficients of automorphic L-functions attached to different cuspidal representations are orthogonal. Professor Jianya Liu and I first proved a weaker, weighted version of this conjecture. As an application, we then proved that if an L-function can be factored into a product of L-functions of possibly different GL(m) over Q, then this factorization is unique. In other words, we proved the uniqueness of functoriality in this case. In particular, an L-function attached to a cuspidal representation of GL(m) over Q cannot be factored further. The proofs are unconditional. Next we proved the original version of Selberg's orthogonality conjecture under the generalized Ramanujan conjecture. Our results can be used to characterize asymptotically whether two cuspidal representations are equivalent, twisted equivalent, or not twisted equivalent at all. This proof also allowed us to study statistical correlations between non trivial zeros of two automorphic L-functions, under the Ramanujan conjecture. Our proofs are unconditional when the representations are corresponding to holomorphic cusp forms for GL(2).

Oct. 8, 5:00pm--6:00pm

Jianya Liu, Subconvexity for Rankin-Selberg L-functions
This is a joint work with Yangbo Ye. We prove a subconvexity bound for Rankin-Selberg $L$-functions $L(s,f\otimes g)$ associated with a Maass cusp form $f$ and a fixed cusp form $g$ in the aspect of the Laplace eigenvalue $1/4+k^2$ of $f$, on the critical line $\text{Re}s=1/2$. Using this subconvexity bound, we prove the equidistribution conjecture of Rudnick and Sarnak on quantum unique ergodicity for dihedral Maass forms. Also proved here is that the generalized Lindel\"{o}f hypothesis for the central value of our $L$-function is true on average.

Oct. 22, 4:00pm--5:00pm

Song Wang, A Cuspidality Criterion for the Fucntiorial Product on GL(2)xGL(3), with a Cohomological Application
A recent breakthrough in automorphic forms is made by H. Kim and F. Shahidi as they established the automorphy of the functorial product on GL(2)xGL(3). However, they did not give any criterion that tells whether such functorial product is cuspidal. The first example of cuspidal such product was given by me earlier in \cite{W}, and also a partial cuspidality criterion was given there. The full version of the cuspidality criterion was just established by D. Ramakrishnan and I (\cite{RW}). In fact, If \pi' is a nondihedral cusp forms on GL(2) and \pi is a cusp form on GL(3), then $\pi'\boxtimes\pi$, as an automorphic form on GL(6), is cuspidal unless \pi is twist equivalent to Ad (\pi'). In this talk, we will explain this cuspidality criterion, and show how to construct non-selfdual cuspidal cohomology classes for (suitable congruence subgroups of) SL(6,Z) as an application of such cuspidality criterion. The basic idea is the following: Let \pi be a cusp form associated to a non-self dual, non-monomial cohomology class on GL(3), and \pi' be a cusp form associated to a holomorphic new form of weight 4. Then $\pi'\boxtimes\pi$ is a cusp form on GL(6) which gives rise to a cohomology class for some congruence subgroup of SL(6,Z). Moreover, choosing $\pi'$ suitably, such homology class can be shown to be not essentially selfdual, not monomial. Our cuspidality criterion plays a central role in this construction. This talk is based on \cite{RW}, which is my joint paper with D. Ramakrishnan.

Oct. 29, 4:00pm--5:00pm

Stephen Kudla, An Arithmetic Inner Product Formula
In recent joint work with Michael Rapoport and Tonghai Yang, we consider a generating function for the classes in the arithmetic Chow group of certain divisors on the arithmetic surface attached to a Shimura curve. This generating function is a modular form of weight 3/2, a kind of arithmetic theta function. We consider the height pairing of two such generating series. The main result is that this arithmetic inner product coincides with the restriction to the diagonal of a certain Eisenstein series of weight 3/2 and genus 2.

Nov. 12, 4:00pm--5:00pm

Akshay Venkatesh, Arithmetic Quantum Unique Ergodicity on Locally Symmetric Spaces
I shall discuss the large-eigenvalue behavior of automorphic forms, i.e. eigenfunctions on "arithmetic" locally symmetric spaces. In particular, I will discuss some recent work with Lior Silberman on the quantum unique ergodicity conjecture, i.e. equidistribution of mass for such eigenfunctions. If there's time, I'll discuss L^{\infty} norms in this context, and where one might look for unusually large values of eigenfunctions.

Dec. 3, 4:00pm--5:00pm

Jeffery Hoffstein, Multiple Dirichlet Series and Dynkin Diagrams
I'll describe the technique of multiple Dirichlet series and survey some recent applications to number theory and automorphic forms. I'll also explain a curious as yet heuristic connection between "perfect" multiple Dirichlet series and Dynkin diagrams.

Dec. 10, 4:00pm--5:00pm

Stephen Miller, Cancellation in sums with additive twists
My talk will center on bounds for sums of the form a(n) exp(2 pi i n x), n = 1 , ... , T . Here a(n) is an arithmetically-interesting sequence of coefficients, such as the coefficients of an L-function or modular form. When the a(n) have average size 1, the trivial bound for this sum is O(T). For modular forms on GL(2), it is not hard to prove the bound O(T^a), for any real number a>1/2 (which in fact is best possible); however for the general automorphic form/L-function on GL(n) achieving this square-root cancellation is a very deep problem (more or less equivalent to the Lindelof Hypothesis in the t-aspect). I will discuss a recent non-trivial result for GL(n), n > 2 : that when the a(n) are the Fourier coefficients of a cusp form on GL(3,Z), the sum is bounded by O(T^(3/4+epsilon)), for any epsilon > 0. If time permits, I will describe how these cancellation questions and some variants are related to proving the existence of zeroes of L-functions on their critical lines.